Geometry: A Metric Approach with Models

geometry is a very appealing but difficult subject mathematically. it concerns the possible configurations of objects of various dimensions and how they meet (are "incident to") one another within a given universe. there are many ways to study it, but there is no easy way to make that study a rigorous mathematical discipline. it is so complicated that even the great mathematicians of antiquity overlooked some details that are actually very important even in flat plane geometry, not to mention spherical or elliptical geometry. to many of us geometry means the euclidean geometry we studied in school, with its list of axioms and postulates and the theorems that we had to memorize proofs for. This hallowed subject has almost disappeared from schools of today, since it resembled latin or greek too much for current tastes, i.e. one just memorized the facts with no thought as to their relevance or meaning. there were two flaws in this old subject, one it was unmotivated and of interest only to the few. second, the historical treatment actually contained several logically significant gaps, i.e. errors that were discovered a hundred or two years ago. modern books thus usually try to present geometry with more effort to make it appealing, and also with an effort to correct the logical mistakes of the ancient geometers. this is hard to do. the logical errors of the ancients are of course subtle and difficult, since the ancients who overlooked them were no slouches themselves. but geometry has always been a place where logical reasoning was practiced in a fairly simple setting, and this tradition seems valuable. the current book tries to present a logically complete and rigorous treatment of plane geometry that is both modern and clear, and well illustrated with numerous examples. the problem of axiomatizing geometry is the following: suppose we start with an example of a geometry, a physical space with sub spaces that can intersect each other, and possibly with the ability to measure angles and lengths, and we want to begin to deduce new facts about it. We need some basic principles that we can use in our reasoning, some axioms or facts that are guaranteed to be true. At the other extreme, we would like to be able to actually recognize when another geometry is essentially the same as ours by checking a few fundamental properties. So we seek to enunciate some basic facts about our geometry that encode its esential features, and we hope to find enough of these, in order to prove that no other geometry satisfies all of them. then we will have characterized our geometry uniquely. euclid had in mind a plane model of geometry that is essentially the plane R^2 of analytic geometry, except he tried to characterize it "synthetically" by axioms which did not involve length and angle measure. He and his contemporaries agonized over just which axioms were needed to completely describe it. In particular they were unsure whether the "5th parallel pos